Two moments from the classic mockumentary This Is Spinal Tap will surely occur to you while watching this funny, sad, flawed documentary about a casualty of the 1970s Detroit music scene. The first is when Tap are devastated to hear from a local DJ that they reside in the "Where Are They Now?" file; the second is when they ecstatically discover that they are big in Japan.

In 1970, a Mexican-American singer-songwriter called Sixto Rodriguez released an album called Cold Fact, a collection of the poetic and socially engaged songs that he'd been singing around the bars and dives of Detroit, Michigan – songs like "Sugar Man", about drugs. I hadn't heard of him before this film; he's a stylish, mysterious figure in dark glasses, looking a little like José Feliciano. His excitable producers were convinced this would make him as big as Dylan. It didn't. Rodriguez followed it up in 1971 with another album called Coming from Reality, but despite what were apparently decent reviews in the music press, Rodriguez's albums sold badly; he was unceremoniously dropped by his record label and turned to manual labour to make ends meet.

Then a strange thing happened: a bootleg copy of Cold Fact seems to have found its way to apartheid-era South Africa where it became a word-of-mouth hit among the white liberal classes, and his powerful, plangent voice became a soundtrack for the whites' anti-apartheid movement. Scenting a hit, South African record labels picked up distribution rights from Rodriguez's US label, Sussex Records, and shifted something in the region of half a million copies, despite some of its songs being banned on South African radio. One interviewee claims that as far as white South Africans were concerned, Cold Fact was as big as Abbey Road or Bridge Over Troubled Water.

The companies paid royalties to the American company, but none of the money appears to have made its way back to poor Rodriguez, and he didn't hear the news that he was famous there. When challenged on this point, the company's boss Clarence Avant becomes very testy and defensive on camera. (In truth, he may have sold these rights for a song anyway.) But what happened to Rodriguez in the end? His South African fans remained mystified, and some even circulated an urban myth that he took his own life on stage.

There is something fascinating in the reclusive rock star who vanishes. In real life, it is Syd Barrett or Peter Green, in fiction, Bucky Wunderlick in Don DeLillo's novel Great Jones Street, or Richard Katz in Jonathan Franzen's Freedom. But none of these examples exactly matches the extraordinary case of Rodriguez. The truth about him – hard to get at in that pre-internet age, and still not entirely illuminated in this film – turned out to be stranger than fiction.

Why were his records so big in South Africa? The movie doesn't spell it out, but his Latino identity was surely key. Being Mexican, he escaped the white/black divide: liberal South African whites music-lovers could embrace him more easily. Yet showbusiness and the music world are capricious and cruel. Rodriguez may well have deserved as much fame as Dylan, or Leonard Cohen or Joni Mitchell. Or a little bit of fame, at least. But it was not to be. In the 21st century, America's entertainment industries may take global markets much more seriously, but back in the 1970s, being popular outside the US was more of an irrelevance, and not paying artists their foreign royalties, or alerting them to their overseas popularity, was something companies could perhaps get away with more easily. Career management was more chaotic as well.

Here, though, we come to the flaw in the movie. It gives the audience the impression that after Rodriguez was dropped by the label, he simply collapsed into non-showbiz obscurity until his South African fanbase was mobilised. But director Malik Bendjelloul is guilty of the sin of omission. A rudimentary internet search shows that Rodriguez's musical career did not vanish the way the film implies, and the film has clearly skated round some facts, and frankly exaggerated the mystery, to make a better and more emotional story.

Well, the broad thesis holds up, and Searching for Sugar Man is an interesting footnote to a species of secret or denied cultural history: the history of South Africa's white liberal classes, the fabric of whose lives may be overlooked by social historians. Not all whites were philistine racists: many were trapped by history and fate and the apartheid laws they detested. Rodriguez's music gave them a voice – and helped indirectly to change things. This movie might itself make a modest contribution to rewriting the history of white South Africa.